Abstract: Consider any nonelementary action of a hyperbolic group G on a not necessarily proper Gromov hyperbolic space X. The action is not assumed to be discrete (for example, it could be a dense subgroup of SL_{2}\R) and X is not assumed to be proper (for example it could be the curve complex, on which the mapping class group acts with pseudo-Anosov elements acting as loxodromics).

We prove certain asymptotic properties for the action, including the following.

1)With respect to the Patterson-Sullivan measure on the boundary of G, the image in X of almost every word-geodesic in G sublinearly tracks a geodesic in X.

2)The proportion of elements in a Cayley-ball of radius R in G which act loxodromically on X converges to 1 with R.

A major tool is Cannon’s theorem that hyperbolic groups admit geodesic automation.

I will also discuss applications to other Markov processes on groups such as non-backtracking random walks and ideas for extending these methods beyond hyperbolic groups. This is based on completed and ongoing work with Sam Taylor and Giulio Tiozzo.